OK, for conventional classical-music recording (i.e. recorded with microphones placed somewhere near the instrument, in several takes, and subsequently edited to produce the master) I think I can speak with some authority, having recorded, edited and mastered a dozen or two of such discs for commercial release and literally hundreds more for demo purposes.
To deal with the question implied in your title first, I use Cool Edit Pro software - since I bought my copy it has been rebranded as Adobe Audition but the central features are absolutely unchanged. It is just about the most capable package for this kind of work: it isn't always the quickest (though there's not much in it given the most time-consuming part of editing is invariably listening through loads of takes deciding what's in and what's out) and it may not be the easiest to use, though that's a bit of a personal choice of course.
Badly voiced chord - can you correct it electronically, or do you just do a retake and cut and paste it into the soundwave at the relevant point?
In theory one can do a very limited amount of electronic correction, and I have done so in desperate circumstances (i.e the artists have long departed the recording studio and there isn't a single satisfactory take). I have even put in notes that were never there. But it's horribly time-consuming, this kind of thing, and on the whole you're better off with an alternative take that can be spliced in.
Wrong note in a passage where there is pedalling and the wrong note hangs over into the next bar or two - again, retake or can it be corrected?
See above. Actually, taking out wrong notes is extremely difficult under any circumstances - in general you'd better regard it as impossible. I said I'm experienced at recording, and doing stuff that's 'not in the manual' is my particular speciality, but taking out wrong notes is one of the hardest tricks.
Tempo in sample 1 isn't quite the same as the tempo in sample 2 - how do you "Hatto" it
Funnily enough, that's very seldom a problem. Assuming the artist knows the piece well (if not the whole recording is a very chancy operation, obviously!) there is unlikely to be a vast difference between one take and another and the listener is very forgiving of this kind of thing. Recently I did have this problem on one particular edit and I used 'Amazing Slow Downer' (Google it) to tweak one take for a few bars to fix it.
I highly recommend getting someone to produce the recording for you - this basically means listening to the recording as it happens and keeping track of what's good and what needs taking again. Anyone with a good musical ear can do this, frankly, though obviously someone who can read the score and spot wrong notes etc. will be most useful. Self-producing is tricky (again, I've done it a few times) and does require a bit more time listening to playbacks at the session.
I notice you say:
what are good editing packages for manipulating sound samples, taking them apart, putting them together, all that sort of stuff
If you're recording a real piano with real microphones you can't treat it all as sound samples! - and therefore can't take them apart. What you can do is edit (bars 1-5 from Take 1, bar 6 and the first chord of bar 7 from Take 3.... etc. etc.). I hope that's what you meant....
Do you have a nice venue to record in? That's typically the most significant factor in determining sound quality of the finished product. If you want any advice on the recording part, just ask.